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Theorem sspwimpALT2 40681
 Description: If a class is a subclass of another class, then its power class is a subclass of that other class's power class. Left-to-right implication of Exercise 18 of [TakeutiZaring] p. 18. Proof derived by completeusersproof.c from User's Proof in VirtualDeductionProofs.txt. The User's Proof in html format is displayed in https://us.metamath.org/other/completeusersproof/sspwimpaltvd.html. (Contributed by Alan Sare, 11-Sep-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
sspwimpALT2 (𝐴𝐵 → 𝒫 𝐴 ⊆ 𝒫 𝐵)

Proof of Theorem sspwimpALT2
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 vex 3412 . . . 4 𝑥 ∈ V
2 elpwi 4426 . . . . 5 (𝑥 ∈ 𝒫 𝐴𝑥𝐴)
3 id 22 . . . . 5 (𝐴𝐵𝐴𝐵)
42, 3sylan9ssr 3866 . . . 4 ((𝐴𝐵𝑥 ∈ 𝒫 𝐴) → 𝑥𝐵)
5 elpwg 4424 . . . . 5 (𝑥 ∈ V → (𝑥 ∈ 𝒫 𝐵𝑥𝐵))
65biimpar 470 . . . 4 ((𝑥 ∈ V ∧ 𝑥𝐵) → 𝑥 ∈ 𝒫 𝐵)
71, 4, 6sylancr 578 . . 3 ((𝐴𝐵𝑥 ∈ 𝒫 𝐴) → 𝑥 ∈ 𝒫 𝐵)
87ex 405 . 2 (𝐴𝐵 → (𝑥 ∈ 𝒫 𝐴𝑥 ∈ 𝒫 𝐵))
98ssrdv 3858 1 (𝐴𝐵 → 𝒫 𝐴 ⊆ 𝒫 𝐵)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 387   ∈ wcel 2050  Vcvv 3409   ⊆ wss 3823  𝒫 cpw 4416 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-ext 2744 This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-clab 2753  df-cleq 2765  df-clel 2840  df-nfc 2912  df-v 3411  df-in 3830  df-ss 3837  df-pw 4418 This theorem is referenced by: (None)
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